“‘I’ll follow him to the ends of the earth,’ she sobbed. Yes, darling. But the earth doesn’t have any ends.” -Tom Robbins
We have some pretty good definitions of what it takes to be a planet, and one part of that definition is that a world needs to be massive enough to pull itself into hydrostatic equilibrium. In the absence of external forces and rotation, that means it will be a perfect sphere.
The line for a planet vs. a non-planet is mass-dependent, and making a thin, rigid body fails on that account. You can have a flat “thing” in space, but it wouldn’t be a planet if you did. Image credit: Margot (2015), via http://arxiv.org/abs/1507.06300.
But what about if you allow the other forces to come into play? In addition to the many interesting features you’ll get, one of them is a flattening of your world. So that brings up the question of how flat a planet could possibly be? This isn’t just theory; our own Solar System has a great example that you’ll want to see for yourself!
A model of Haumea rotating, based on the most accurate data available. Image credit: Wikimedia Commons user Stephanie Hoover.